Deriving the Area Formula for a Circle

Date: 03/08/2000 at 17:56:38
From: Richard Atkinson
Subject: Area of a circle as it relates to pi
Why is the area of a circle the square of the radius times pi?

Date: 03/09/2000 at 03:23:03
From: Doctor Floor
Subject: Re: Area of a circle as it relates to pi
Hi, Richard,
Thanks for writing.
Let's consider a circle with radius r.
If we divide the circle into an even number of sectors, we can
rearrange these sectors as in the following figure:
The result is a sort of wrongly formed rectangle, but we know that the
shorter "side" of this rectangle has length r, and that the longer
"side" is half the perimeter, hence pi*r.
The more sectors we make, the more accurate our rectangle becomes. We
can imagine that if we divided the circle into an infinite number of
sectors, it would become a rectangle.
Whatever the number of sectors we use, the "side" lengths will remain
r and pi*r. Therefore our limit case with an infinite number of
sectors still has sides r and pi*r, and the area of this limit
rectangle is pi*r^2. Since the area of the circle does not change when
we divide it into parts, the area of the circle must have been pi*r^2,
too.
If you need more help, just write back.
Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/